This paper presents efficient data structures for the implementation of matrix-free finite element methods on block-structured, hybrid tetrahedral grids. It provides a complete categorization of all geometric sub-objects that emerge from the regular refinement of the unstructured, tetrahedral coarse grid and describes efficient iteration patterns and analytical linearization functions for the mapping of coefficients to memory addresses. This foundation enables the implementation of fast, extreme-scalable, matrix-free, iterative solvers, and in particular geometric multigrid methods by design. Their application to the variable-coefficient Stokes system subject to an enriched Galerkin discretization and to the curl-curl problem discretized with N\'ed\'elec edge elements showcases the flexibility of the implementation. Eventually, the solution of a curl-curl problem with $1.6 \cdot 10^{11}$ (more than one hundred billion) unknowns on more than $32000$ processes with a matrix-free full multigrid solver demonstrates its extreme-scalability.

翻译：本文提出了高效的数据结构，用于在块结构混合四面体网格上实现无矩阵有限元方法。它完整分类了通过非结构四面体粗网格正则加密产生的所有几何子对象，描述了高效的迭代模式及用于系数到内存地址映射的解析线性化函数。这一基础为实现快速、极端可扩展、无矩阵、迭代求解器（特别是几何多重网格方法）提供了设计支撑。将其应用于富集伽辽金离散的可变系数斯托克斯系统以及用Nédélec边元离散的旋度-旋度问题，展示了实现的灵活性。最终，在超过32000个进程上求解具有1.6·10¹¹（超过一千亿）未知量的旋度-旋度问题，通过无矩阵全多重网格求解器验证了其极端可扩展性。